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Nov 27, 2025
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Understanding Probability: Key Concepts and Equations
E
Emma Bronkar
@ebronkar
1 / 4
Understanding Probability Basics
Ever wondered what your chances are of winning a game? Probability gives us the answer! Probability is the proportion of times an outcome would occur if you observed a random process an infinite number of times. It's always expressed as a number between 0 (impossible) and 1 (certain).
When calculating probability, we need to understand the sample space - all possible outcomes of a random process. For example, when rolling a six-sided die, the sample space is {1, 2, 3, 4, 5, 6}. The probability of rolling a 4 is 1/6 or approximately 0.167, because there's one favorable outcome out of six possible outcomes.
To find probability mathematically, we use the formula P(A) = n(A)/n(sample space). This means the probability of event A equals the number of ways A can happen divided by the total number of possible outcomes. Remember that the sum of all probabilities in a sample space must equal 1, which represents 100% certainty that something from the sample space will occur.
💡 Quick Check: If you calculate a probability less than 0 or greater than 1, you've made an error! Probabilities always fall within the range 0 ≤ P(A) ≤ 1, because events can only be between 0% and 100% likely to happen.
Compound Probabilities
The Law of Large Numbers explains why probability works in real life - as you repeat a random process more times, the actual results get closer to the theoretical probability. If you flip a fair coin 10 times, getting 7 heads isn't unusual, but flipping 700 heads in 1000 tosses would be surprising!
When calculating probabilities of multiple events, we use different approaches. For "and" probability (when multiple events need to happen together), we multiply the individual probabilities. For independent events: P(A and B) = P(A) × P(B). For example, the probability of rolling a 4 and then a 5 on a die equals (1/6) × (1/6) = 1/36 or about 0.028.
For "or" probability (when either event A, event B, or both happen), we use: P(A or B) = P(A) + P(B) - P(A and B). This subtraction prevents double-counting events that satisfy both conditions. For instance, if 20% of students are left-handed and 60% have brown eyes, with 11.5% having both traits, the probability of randomly selecting a student who is either left-handed or brown-eyed is 0.2 + 0.6 - 0.115 = 0.685.
💡 Remember: "Or" probability includes situations where both events happen! That's why we need to subtract the overlap (the "and" probability) to avoid counting those instances twice.
Conditional Probability and Event Relationships
Sometimes events can't happen together - these are called disjoint outcomes or mutually exclusive events. For example, a single coin flip can't be both heads and tails. When events are disjoint, P(A and B) = 0, which simplifies our "or" probability formula to P(A or B) = P(A) + P(B).
Conditional probability helps us calculate the likelihood of an event given that another event has already occurred. We write this as P(B|A), meaning "probability of B given that A has happened." The formula is P(B|A) = P(A and B)/P(A). For example, if 30% of students have blue eyes, 60% have brown hair, and 10% have both, then the probability of a brown-haired student having blue eyes is 0.1/0.6 = 0.167.
Events can be either independent or dependent. Independent events don't affect each other's probabilities - like consecutive coin flips where P(B|A) = P(B). Dependent events, however, do influence each other - like drawing cards without replacement. For dependent events, the probability changes after each event occurs.
💡 Pro Tip: When solving dependent event problems, remember to update your sample space and favorable outcomes after each event. For drawing two red marbles without replacement from a bag with 8 red marbles out of 20, the probability would be (8/20) × (7/19) = 0.147.
The 10% Rule and Practical Applications
The 10% rule simplifies probability calculations in certain situations. When sampling from a very large population (where the sample size is less than 10% of the population), we can treat events as independent even if technically they're not. This approximation saves time without significantly affecting accuracy.
For example, if there are 1 billion coins with 400 million being pennies, the probability of drawing 5 pennies in a row would be approximately (0.4)^5 = 0.01024, even though each draw slightly changes the composition of the remaining coins. Because the sample (5 coins) is minuscule compared to the population (1 billion), this approximation works well.
This rule has important practical applications in surveys and statistics. When polling voters in a country of millions, statisticians can treat each selection as independent as long as the sample remains small relative to the population, making calculations much more manageable.
💡 Simplification Tip: When working with very large populations, use the 10% rule to treat dependent events as independent. If your sample is less than 10% of the population, the simplified calculation will be close enough for practical purposes!
We thought you’d never ask...
What is the Knowunity AI companion?
Our AI companion is specifically built for the needs of students. Based on the millions of content pieces we have on the platform we can provide truly meaningful and relevant answers to students. But its not only about answers, the companion is even more about guiding students through their daily learning challenges, with personalised study plans, quizzes or content pieces in the chat and 100% personalisation based on the students skills and developments.
Where can I download the Knowunity app?
You can download the app in the Google Play Store and in the Apple App Store.
Is Knowunity really free of charge?
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4.8/5
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The app is very easy to use and well designed. I have found everything I was looking for so far and have been able to learn a lot from the presentations! I will definitely use the app for a class assignment! And of course it also helps a lot as an inspiration.
Stefan S
iOS user
This app is really great. There are so many study notes and help [...]. My problem subject is French, for example, and the app has so many options for help. Thanks to this app, I have improved my French. I would recommend it to anyone.
Samantha Klich
Android user
Wow, I am really amazed. I just tried the app because I've seen it advertised many times and was absolutely stunned. This app is THE HELP you want for school and above all, it offers so many things, such as workouts and fact sheets, which have been VERY helpful to me personally.
Anna
iOS user
I think it’s very much worth it and you’ll end up using it a lot once you get the hang of it and even after looking at others notes you can still ask your Artificial intelligence buddy the question and ask to simplify it if you still don’t get it!!! In the end I think it’s worth it 😊👍 ⚠️Also DID I MENTION ITS FREEE YOU DON’T HAVE TO PAY FOR ANYTHING AND STILL GET YOUR GRADES IN PERFECTLY❗️❗️⚠️
Thomas R
iOS user
Knowunity is the BEST app I’ve used in a minute. This is not an ai review or anything this is genuinely coming from a 7th grade student (I know 2011 im young) but dude this app is a 10/10 i have maintained a 3.8 gpa and have plenty of time for gaming. I love it and my mom is just happy I got good grades
Brad T
Android user
Not only did it help me find the answer but it also showed me alternative ways to solve it. I was horrible in math and science but now I have an a in both subjects. Thanks for the help🤍🤍
David K
iOS user
The app's just great! All I have to do is enter the topic in the search bar and I get the response real fast. I don't have to watch 10 YouTube videos to understand something, so I'm saving my time. Highly recommended!
Sudenaz Ocak
Android user
In school I was really bad at maths but thanks to the app, I am doing better now. I am so grateful that you made the app.
Greenlight Bonnie
Android user
I found this app a couple years ago and it has only gotten better since then. I really love it because it can help with written questions and photo questions. Also, it can find study guides that other people have made as well as flashcard sets and practice tests. The free version is also amazing for students who might not be able to afford it. Would 100% recommend
Aubrey
iOS user
Best app if you're in Highschool or Junior high. I have been using this app for 2 school years and it's the best, it's good if you don't have anyone to help you with school work.😋🩷🎀
Marco B
iOS user
THE QUIZES AND FLASHCARDS ARE SO USEFUL AND I LOVE THE SCHOOLGPT. IT ALSO IS LITREALLY LIKE CHATGPT BUT SMARTER!! HELPED ME WITH MY MASCARA PROBLEMS TOO!! AS WELL AS MY REAL SUBJECTS ! DUHHH 😍😁😲🤑💗✨🎀😮
Elisha
iOS user
This app is phenomenal down to the correct info and the various topics you can study! I greatly recommend it for people who struggle with procrastination and those who need homework help. It has been perfectly accurate for world 1 history as far as I’ve seen! Geometry too!
Paul T
iOS user
The app is very easy to use and well designed. I have found everything I was looking for so far and have been able to learn a lot from the presentations! I will definitely use the app for a class assignment! And of course it also helps a lot as an inspiration.
Stefan S
iOS user
This app is really great. There are so many study notes and help [...]. My problem subject is French, for example, and the app has so many options for help. Thanks to this app, I have improved my French. I would recommend it to anyone.
Samantha Klich
Android user
Wow, I am really amazed. I just tried the app because I've seen it advertised many times and was absolutely stunned. This app is THE HELP you want for school and above all, it offers so many things, such as workouts and fact sheets, which have been VERY helpful to me personally.
Anna
iOS user
I think it’s very much worth it and you’ll end up using it a lot once you get the hang of it and even after looking at others notes you can still ask your Artificial intelligence buddy the question and ask to simplify it if you still don’t get it!!! In the end I think it’s worth it 😊👍 ⚠️Also DID I MENTION ITS FREEE YOU DON’T HAVE TO PAY FOR ANYTHING AND STILL GET YOUR GRADES IN PERFECTLY❗️❗️⚠️
Thomas R
iOS user
Knowunity is the BEST app I’ve used in a minute. This is not an ai review or anything this is genuinely coming from a 7th grade student (I know 2011 im young) but dude this app is a 10/10 i have maintained a 3.8 gpa and have plenty of time for gaming. I love it and my mom is just happy I got good grades
Brad T
Android user
Not only did it help me find the answer but it also showed me alternative ways to solve it. I was horrible in math and science but now I have an a in both subjects. Thanks for the help🤍🤍
David K
iOS user
The app's just great! All I have to do is enter the topic in the search bar and I get the response real fast. I don't have to watch 10 YouTube videos to understand something, so I'm saving my time. Highly recommended!
Sudenaz Ocak
Android user
In school I was really bad at maths but thanks to the app, I am doing better now. I am so grateful that you made the app.
Greenlight Bonnie
Android user
I found this app a couple years ago and it has only gotten better since then. I really love it because it can help with written questions and photo questions. Also, it can find study guides that other people have made as well as flashcard sets and practice tests. The free version is also amazing for students who might not be able to afford it. Would 100% recommend
Aubrey
iOS user
Best app if you're in Highschool or Junior high. I have been using this app for 2 school years and it's the best, it's good if you don't have anyone to help you with school work.😋🩷🎀
Marco B
iOS user
THE QUIZES AND FLASHCARDS ARE SO USEFUL AND I LOVE THE SCHOOLGPT. IT ALSO IS LITREALLY LIKE CHATGPT BUT SMARTER!! HELPED ME WITH MY MASCARA PROBLEMS TOO!! AS WELL AS MY REAL SUBJECTS ! DUHHH 😍😁😲🤑💗✨🎀😮
Elisha
iOS user
This app is phenomenal down to the correct info and the various topics you can study! I greatly recommend it for people who struggle with procrastination and those who need homework help. It has been perfectly accurate for world 1 history as far as I’ve seen! Geometry too!
Paul T
iOS user
Understanding Probability: Key Concepts and Equations
E
Emma Bronkar
@ebronkar
Probability is all about measuring the likelihood of events in random processes. As you'll discover, this mathematical concept helps us predict outcomes based on known information. Whether it's calculating the odds of drawing specific cards, rolling certain numbers, or predicting... Show more
Understanding Probability Basics
Ever wondered what your chances are of winning a game? Probability gives us the answer! Probability is the proportion of times an outcome would occur if you observed a random process an infinite number of times. It's always expressed as a number between 0 (impossible) and 1 (certain).
When calculating probability, we need to understand the sample space - all possible outcomes of a random process. For example, when rolling a six-sided die, the sample space is {1, 2, 3, 4, 5, 6}. The probability of rolling a 4 is 1/6 or approximately 0.167, because there's one favorable outcome out of six possible outcomes.
To find probability mathematically, we use the formula P(A) = n(A)/n(sample space). This means the probability of event A equals the number of ways A can happen divided by the total number of possible outcomes. Remember that the sum of all probabilities in a sample space must equal 1, which represents 100% certainty that something from the sample space will occur.
💡 Quick Check: If you calculate a probability less than 0 or greater than 1, you've made an error! Probabilities always fall within the range 0 ≤ P(A) ≤ 1, because events can only be between 0% and 100% likely to happen.
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Compound Probabilities
The Law of Large Numbers explains why probability works in real life - as you repeat a random process more times, the actual results get closer to the theoretical probability. If you flip a fair coin 10 times, getting 7 heads isn't unusual, but flipping 700 heads in 1000 tosses would be surprising!
When calculating probabilities of multiple events, we use different approaches. For "and" probability (when multiple events need to happen together), we multiply the individual probabilities. For independent events: P(A and B) = P(A) × P(B). For example, the probability of rolling a 4 and then a 5 on a die equals (1/6) × (1/6) = 1/36 or about 0.028.
For "or" probability (when either event A, event B, or both happen), we use: P(A or B) = P(A) + P(B) - P(A and B). This subtraction prevents double-counting events that satisfy both conditions. For instance, if 20% of students are left-handed and 60% have brown eyes, with 11.5% having both traits, the probability of randomly selecting a student who is either left-handed or brown-eyed is 0.2 + 0.6 - 0.115 = 0.685.
💡 Remember: "Or" probability includes situations where both events happen! That's why we need to subtract the overlap (the "and" probability) to avoid counting those instances twice.
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Conditional Probability and Event Relationships
Sometimes events can't happen together - these are called disjoint outcomes or mutually exclusive events. For example, a single coin flip can't be both heads and tails. When events are disjoint, P(A and B) = 0, which simplifies our "or" probability formula to P(A or B) = P(A) + P(B).
Conditional probability helps us calculate the likelihood of an event given that another event has already occurred. We write this as P(B|A), meaning "probability of B given that A has happened." The formula is P(B|A) = P(A and B)/P(A). For example, if 30% of students have blue eyes, 60% have brown hair, and 10% have both, then the probability of a brown-haired student having blue eyes is 0.1/0.6 = 0.167.
Events can be either independent or dependent. Independent events don't affect each other's probabilities - like consecutive coin flips where P(B|A) = P(B). Dependent events, however, do influence each other - like drawing cards without replacement. For dependent events, the probability changes after each event occurs.
💡 Pro Tip: When solving dependent event problems, remember to update your sample space and favorable outcomes after each event. For drawing two red marbles without replacement from a bag with 8 red marbles out of 20, the probability would be (8/20) × (7/19) = 0.147.
Sign up to see the contentIt's free!
Access to all documents
Improve your grades
Join milions of students
By signing up you accept Terms of Service and Privacy Policy
The 10% Rule and Practical Applications
The 10% rule simplifies probability calculations in certain situations. When sampling from a very large population (where the sample size is less than 10% of the population), we can treat events as independent even if technically they're not. This approximation saves time without significantly affecting accuracy.
For example, if there are 1 billion coins with 400 million being pennies, the probability of drawing 5 pennies in a row would be approximately (0.4)^5 = 0.01024, even though each draw slightly changes the composition of the remaining coins. Because the sample (5 coins) is minuscule compared to the population (1 billion), this approximation works well.
This rule has important practical applications in surveys and statistics. When polling voters in a country of millions, statisticians can treat each selection as independent as long as the sample remains small relative to the population, making calculations much more manageable.
💡 Simplification Tip: When working with very large populations, use the 10% rule to treat dependent events as independent. If your sample is less than 10% of the population, the simplified calculation will be close enough for practical purposes!
We thought you’d never ask...
What is the Knowunity AI companion?
Our AI companion is specifically built for the needs of students. Based on the millions of content pieces we have on the platform we can provide truly meaningful and relevant answers to students. But its not only about answers, the companion is even more about guiding students through their daily learning challenges, with personalised study plans, quizzes or content pieces in the chat and 100% personalisation based on the students skills and developments.
Where can I download the Knowunity app?
You can download the app in the Google Play Store and in the Apple App Store.
Is Knowunity really free of charge?
That's right! Enjoy free access to study content, connect with fellow students, and get instant help – all at your fingertips.
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App Store
4.8/5
Google Play
The app is very easy to use and well designed. I have found everything I was looking for so far and have been able to learn a lot from the presentations! I will definitely use the app for a class assignment! And of course it also helps a lot as an inspiration.
Stefan S
iOS user
This app is really great. There are so many study notes and help [...]. My problem subject is French, for example, and the app has so many options for help. Thanks to this app, I have improved my French. I would recommend it to anyone.
Samantha Klich
Android user
Wow, I am really amazed. I just tried the app because I've seen it advertised many times and was absolutely stunned. This app is THE HELP you want for school and above all, it offers so many things, such as workouts and fact sheets, which have been VERY helpful to me personally.
Anna
iOS user
I think it’s very much worth it and you’ll end up using it a lot once you get the hang of it and even after looking at others notes you can still ask your Artificial intelligence buddy the question and ask to simplify it if you still don’t get it!!! In the end I think it’s worth it 😊👍 ⚠️Also DID I MENTION ITS FREEE YOU DON’T HAVE TO PAY FOR ANYTHING AND STILL GET YOUR GRADES IN PERFECTLY❗️❗️⚠️
Thomas R
iOS user
Knowunity is the BEST app I’ve used in a minute. This is not an ai review or anything this is genuinely coming from a 7th grade student (I know 2011 im young) but dude this app is a 10/10 i have maintained a 3.8 gpa and have plenty of time for gaming. I love it and my mom is just happy I got good grades
Brad T
Android user
Not only did it help me find the answer but it also showed me alternative ways to solve it. I was horrible in math and science but now I have an a in both subjects. Thanks for the help🤍🤍
David K
iOS user
The app's just great! All I have to do is enter the topic in the search bar and I get the response real fast. I don't have to watch 10 YouTube videos to understand something, so I'm saving my time. Highly recommended!
Sudenaz Ocak
Android user
In school I was really bad at maths but thanks to the app, I am doing better now. I am so grateful that you made the app.
Greenlight Bonnie
Android user
I found this app a couple years ago and it has only gotten better since then. I really love it because it can help with written questions and photo questions. Also, it can find study guides that other people have made as well as flashcard sets and practice tests. The free version is also amazing for students who might not be able to afford it. Would 100% recommend
Aubrey
iOS user
Best app if you're in Highschool or Junior high. I have been using this app for 2 school years and it's the best, it's good if you don't have anyone to help you with school work.😋🩷🎀
Marco B
iOS user
THE QUIZES AND FLASHCARDS ARE SO USEFUL AND I LOVE THE SCHOOLGPT. IT ALSO IS LITREALLY LIKE CHATGPT BUT SMARTER!! HELPED ME WITH MY MASCARA PROBLEMS TOO!! AS WELL AS MY REAL SUBJECTS ! DUHHH 😍😁😲🤑💗✨🎀😮
Elisha
iOS user
This app is phenomenal down to the correct info and the various topics you can study! I greatly recommend it for people who struggle with procrastination and those who need homework help. It has been perfectly accurate for world 1 history as far as I’ve seen! Geometry too!
Paul T
iOS user
The app is very easy to use and well designed. I have found everything I was looking for so far and have been able to learn a lot from the presentations! I will definitely use the app for a class assignment! And of course it also helps a lot as an inspiration.
Stefan S
iOS user
This app is really great. There are so many study notes and help [...]. My problem subject is French, for example, and the app has so many options for help. Thanks to this app, I have improved my French. I would recommend it to anyone.
Samantha Klich
Android user
Wow, I am really amazed. I just tried the app because I've seen it advertised many times and was absolutely stunned. This app is THE HELP you want for school and above all, it offers so many things, such as workouts and fact sheets, which have been VERY helpful to me personally.
Anna
iOS user
I think it’s very much worth it and you’ll end up using it a lot once you get the hang of it and even after looking at others notes you can still ask your Artificial intelligence buddy the question and ask to simplify it if you still don’t get it!!! In the end I think it’s worth it 😊👍 ⚠️Also DID I MENTION ITS FREEE YOU DON’T HAVE TO PAY FOR ANYTHING AND STILL GET YOUR GRADES IN PERFECTLY❗️❗️⚠️
Thomas R
iOS user
Knowunity is the BEST app I’ve used in a minute. This is not an ai review or anything this is genuinely coming from a 7th grade student (I know 2011 im young) but dude this app is a 10/10 i have maintained a 3.8 gpa and have plenty of time for gaming. I love it and my mom is just happy I got good grades
Brad T
Android user
Not only did it help me find the answer but it also showed me alternative ways to solve it. I was horrible in math and science but now I have an a in both subjects. Thanks for the help🤍🤍
David K
iOS user
The app's just great! All I have to do is enter the topic in the search bar and I get the response real fast. I don't have to watch 10 YouTube videos to understand something, so I'm saving my time. Highly recommended!
Sudenaz Ocak
Android user
In school I was really bad at maths but thanks to the app, I am doing better now. I am so grateful that you made the app.
Greenlight Bonnie
Android user
I found this app a couple years ago and it has only gotten better since then. I really love it because it can help with written questions and photo questions. Also, it can find study guides that other people have made as well as flashcard sets and practice tests. The free version is also amazing for students who might not be able to afford it. Would 100% recommend
Aubrey
iOS user
Best app if you're in Highschool or Junior high. I have been using this app for 2 school years and it's the best, it's good if you don't have anyone to help you with school work.😋🩷🎀
Marco B
iOS user
THE QUIZES AND FLASHCARDS ARE SO USEFUL AND I LOVE THE SCHOOLGPT. IT ALSO IS LITREALLY LIKE CHATGPT BUT SMARTER!! HELPED ME WITH MY MASCARA PROBLEMS TOO!! AS WELL AS MY REAL SUBJECTS ! DUHHH 😍😁😲🤑💗✨🎀😮
Elisha
iOS user
This app is phenomenal down to the correct info and the various topics you can study! I greatly recommend it for people who struggle with procrastination and those who need homework help. It has been perfectly accurate for world 1 history as far as I’ve seen! Geometry too!
Paul T
iOS user